Implied Volatility





Kerry Back

  • The inputs to the Black-Scholes formulas are: strike, time to maturity, interest rate, and volatility (and dividends).
  • All of these are known except volatility (and dividends, which is easy to estimate).
  • We could input historical volatility, but volatility changes over time and the market is forward looking.
  • A standard thing is to infer volatility from market prices of actively traded options. This is called the implied volatility.

from scipy.optimize import fsolve
def callIV(C, S, K, T, r, q=0):
  return fsolve(
    lambda x: C - callBS(S, K, T, x, r, q),
    x0=0.3
  )

C = 15
S = 100
K = 100
T = 1
r = 0.05
iv = callIV(C, S, K, T, r)
iv[0]
0.32025830955048284

AAPL Call Data on 3/6/2023

  • last AAPL price:
153.98
  • April 21, 2023 calls
contractSymbol lastTradeDate lastPrice bid ask change percentChange volume openInterest impliedVolatility inTheMoney contractSize currency
strike
140.0 AAPL230421C00140000 2023-03-06 19:47:20+00:00 16.50 16.50 16.70 2.67 19.305859 533 15042 0.371100 True REGULAR USD
145.0 AAPL230421C00145000 2023-03-06 19:44:08+00:00 12.50 12.45 12.60 2.60 26.262630 1278 11887 0.338752 True REGULAR USD
150.0 AAPL230421C00150000 2023-03-06 19:42:03+00:00 8.70 8.75 8.85 2.00 29.850748 6233 35170 0.306892 True REGULAR USD
155.0 AAPL230421C00155000 2023-03-06 19:45:57+00:00 5.78 5.75 5.80 1.69 41.320293 9023 44191 0.284919 False REGULAR USD
160.0 AAPL230421C00160000 2023-03-06 19:45:34+00:00 3.40 3.40 3.45 1.20 54.545456 9097 43762 0.266487 False REGULAR USD
165.0 AAPL230421C00165000 2023-03-06 19:47:25+00:00 1.74 1.77 1.79 0.68 64.150955 7850 27696 0.248909 False REGULAR USD
170.0 AAPL230421C00170000 2023-03-06 19:45:02+00:00 0.84 0.83 0.85 0.38 82.608690 9391 39456 0.238655 False REGULAR USD

AAPL Call and Put Implied Vols

Call IV Put IV
strike
140.0 0.371100 0.275886
145.0 0.338752 0.258674
150.0 0.306892 0.241707
155.0 0.284919 0.225227
160.0 0.266487 0.203987
165.0 0.248909 0.175423
170.0 0.238655 0.000010

TSLA Call and Put Implied Vols

Call IV Put IV
strike
180.00 0.681400 0.624638
183.33 0.674930 0.619694
185.00 0.670536 0.616215
190.00 0.659549 0.608769
191.67 0.659122 0.608036
195.00 0.654667 0.601200
200.00 0.648441 0.595280
205.00 0.642765 0.591923
208.33 0.640323 0.585026
210.00 0.637028 0.583256

VIX (Volatility Index)

  • average of implied vols of out-of-the-money 1-month-to-maturity S&P 500 put and call options